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Angles This article uses Greek letters such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and grads: 1 full circle = 360 degrees = 2\pi radians = 400 grads. The following table shows the conversions for some common angles: Degrees 30° 60° 120° 150° 210° 240° 300° 330° Radians \frac\pi6\! \frac\pi3\! \frac{2\pi}3\! \frac{5\pi}6\! \frac{7\pi}6\! \frac{4\pi}3\! \frac{5\pi}3\! \frac{11\pi}6\! Grads 33⅓ grad 66⅔ grad 133⅓ grad 166⅔ grad 233⅓ grad 266⅔ grad 333⅓ grad 366⅔ grad Degrees 45° 90° 135° 180° 225° 270° 315° 360° Radians \frac\pi4\! \frac\pi2\! \frac{3\pi}4\! \pi\! \frac{5\pi}4\! \frac{3\pi}2\! \frac{7\pi}4\! 2\pi\! Grads 50 grad 100 grad 150 grad 200 grad 250 grad 300 grad 350 grad 400 grad Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per Niven's theorem multiples of 30° are the only angles that are a rational multiple of one degree and also have a rational sin/cos, which may account for their popularity in examples.[1] Trigonometric functions The primary trigonometric functions are the sine and cosine of an angle. These are sometimes abbreviated sin(θ) and cos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ. The Sine of an angle is defined in the context of a Right Triangle, as the ratio of the length of the side that is opposite to the angle, divided by the length of the longest side of the triangle (the Hypotenuse ). The Cosine of an angle is also defined in the context of a Right Triangle, as the ratio of the length of the side the angle is in, divided by the length of the longest side of the triangle (the Hypotenuse ). The tangent (tan) of an angle is the ratio of the sine to the cosine: \tan\theta = \frac{\sin\theta}{\cos\theta}. Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent: \sec\theta = \frac{1}{\cos\theta},\quad\csc\theta = \frac{1}{\sin\theta},\quad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}. These definitions are sometimes referred to as ratio identities. |
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